The Cohen-Macaulayness of the bounded complex of an affine oriented matroid

TL;DR

This paper explores the Cohen-Macaulay property of affine oriented matroids, linking algebraic conditions to topological properties of their bounded complexes, and characterizes when these complexes are Cohen-Macaulay.

Contribution

It establishes a characterization of affine oriented matroids with Cohen-Macaulay ideals and describes the canonical module in such cases.

Findings

Cohen-Macaulay ideals imply the bounded complex is a contractible homology manifold with boundary.

A complete characterization of affine oriented matroids with Cohen-Macaulay ideals.

Description of the canonical module in the Cohen-Macaulay case.

Abstract

An affine oriented matroid is a combinatorial abstraction of an affine hyperplane arrangement. From it, Novik, Postnikov and Sturmfels constructed a squarefree monomial ideal in a polynomial ring, called an oriented matroid ideal, and got beautiful results. Developing their theory, we will show the following. (1) If an oriented matroid ideal is Cohen-Macaulay, then the bounded complex (a regular CW complex associated with it) of the corresponding affine oriented matroid is a contractible homology manifold with boundary. This is closely related to Dong's theorem, which used to be "Zaslavsky's conjecture". (2) We characterize the affine oriented matroid whose corresponding ideal is Cohen-Macaulay. (3) In the Cohen-Macaulay case, we give a description of the canonical module of the residue class ring by an oriented matroid ideal.